Elsevier

Economics Letters

Volume 196, November 2020, 109547
Economics Letters

Mere Addition is equivalent to avoiding the Sadistic Conclusion in all plausible variable-population social orderings

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Highlights

  • Given a set of weak axioms, avoiding the Sadistic Conclusion implies Mere Addition.

  • One of these axioms says if adding lives is bad, adding more such lives is worse.

  • All plausible orderings that satisfy Mere Addition also satisfy Negative Expansion.

  • So Mere Addition is avoidance of the Sadistic Conclusion for any plausible ordering.

  • These results imply a novel restatement of Ng’s (1989) impossibility theorem.

Abstract

Economic policy evaluations require social welfare functions for variable-size populations. Two important axioms in the population ethics literature are Mere Addition and avoidance of the Sadistic Conclusion, both of which focus on the sign of lifetime utility. The population ethics literature treats these axioms as closely related but distinct: one influential review calls avoidance of the Sadistic Conclusion “less controversial.” Here, we provide weak, uncontroversial sufficient conditions for these two principles to be equivalent. Related results exist in prior literature, but these include only same-number utilitarian orderings and therefore exclude recent and theoretically important rank-dependent social evaluations that we include.

Introduction

Population ethics studies the axiomatic properties of social orderings of welfare vectors with different population sizes. Two well-studied axioms in population ethics hinge on the sign of lifetime utilities. Mere Addition, introduced by Parfit (1984), holds that adding a positive-utility life to any population does not result in a worse population. The Sadistic Conclusion, introduced by Arrhenius (2000), holds that adding negative lives can be strictly preferred to adding positive lives. The literature typically describes the Sadistic Conclusion as ethically worse than denying Mere Addition. Arrhenius (2017) calls the requirement to avoid the Sadistic Conclusion a “less controversial assumption” than Mere Addition (p. 297).

Arrhenius (2017) notes that social orderings that violate Mere Addition “tend to imply” the Sadistic Conclusion (p. 94). Greaves (2017) calls these axioms “related”. Bossert (2017) especially highlights the normative link but less generally than we do. We show the link is stronger than tendency. Under a set of basic plausibility axioms not violated by any social ordering defended in the economics literature, these two conditions are equivalent. In their theorem 5.4, Blackorby et al. (2005) state a related result3 that only applies to social orderings that are same-number generalized utilitarian (e.g., not rank-dependent evaluations, an important recent innovation by Asheim and Zuber, 2014).

Section snippets

Setting and basic axioms

Our notation follows Blackorby et al. (2005). Z is the integers, R is the real numbers, R++ and R+ are the positive and nonnegative real numbers, respectively, and similarly for , , and Z.

Populations u,v are finite-length vectors of real numbers, where the ith position in the vector ui is the lifetime utility of person i. Utilities are normalized so that ui=0 is a neutral life (as good as a life with no experiences) for person i. Following Asheim and Zuber (2014), an index enclosed in square

Mere addition and the Sadistic conclusion

Axiom Mere Addition

For all uΩ and all vΩ++, uvu.

Axiom Avoidance of the Sadistic Conclusion

For all uΩ, all vΩ++, and all wΩ, uvuw.

Total Utilitarianism (W(u)=iui) and Total Prioritarianism (W(u)=igui for increasing, concave g such that g(0)=0) satisfy Mere Addition and avoid the Sadistic Conclusion. Average Utilitarianism (W(u)=1n(u)iui), Average Prioritarianism (W(u)=1n(u)igui), Number-Dampened Utilitarianism (W(u)=1n(u)iuin(u)α, where 0<α<1, yielding a positive, increasing, concave transformation of population size; abbreviated NDGU),

Result

Axiom Consistent Expansion

For all uΩ, vR, and nZ++, if uuv1n then there exists m>n such that uv1nuv1m.

Consistent Expansion holds that if adding nv-lives to u makes a worse combined population, then adding some further number of v-lives makes an even worse population.4 Consistent Expansion is satisfied by Total and Average Utilitarianism and Prioritarianism,

Discussion

Ng’s (1989) classic impossibility theorem shows that no social ordering can:

  • satisfy Mere Addition,

  • satisfy a normatively unobjectionable principle called Non-Antiegalitarianism, and

  • avoid Parfit’s (1984) Repugnant Conclusion.

Because the economics literature has never entertained violating Non-Antiegalitarianism, in practice one must reject Mere Addition or accept Parfit’s Repugnant Conclusion. One implication of our result is that, in the context of the specified axioms, satisfying Mere Addition

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Although this paper received no specific funding, both authors’ research is supported by grant NICHD, United States of America grants K01HD098313 and P2CHD042849. The content is solely the responsibility of the author and does not necessarily represent the official views of the National Institutes of Health.

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